does number adding order affects to the final sum? when the number of numbers are finite, obviously final sum will not depend on the order they are adding(the commutative law). but when the number of numbers are infinite, does the final summation depend on the order of adding or are there any infinite series that are summed up to different numbers when the order of adding is changed? I can't find any useful information by googling.
If I elaborate by examples.

(2+3)+5 = (5+3)+2 is commutative law of adding.

is this property hold for infinite number of numbers or is there any infinite number series that is violate this property. assume these series are summed up to finite summations.
 A: If the series is only conditionally convergent ($\sum a_n$ converges but $\sum |a_n|$ diverges), then rearranging the terms can change the sum.  In fact, the Riemann rearrangement theorem says that a conditionally convergent series (of real numbers) may be rearranged to converge to any real number, to diverge to $\infty$ or to $-\infty$.
However, if the series is absolutely convergent ($\sum |a_n|$ converges), then rearranging or regrouping the terms will not change the sum.
A: If the numbers are all non-negative, the order of summation is irrelevant. 
Otherwise you have to consider the sum of the positive numbers and the sum of the negative numbers. If both of them are infinite, you cannot "sum" the numbers, since changing the order may let you have different results; otherwise you may change the order of summation.
But I believe this should have been easily found by googling.
A: If you are working
in finite precision floating point,
the order of computation
can make a big difference.
For example,
if x is very large
and y is very small,
you can get
(x+y)-x = 0
and
(x-x)+y = y.
